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To solve the quadratic equation 2x - 6 = 4x², we can follow these steps:
1. Subtract 2x from both sides:
2x - 6 - 2x = 4x² - 2x
-6 = 2x²
2. Divide both sides by 2:
-6/2 = 2x²/2
-3 = x²
3. Take the square root of both sides:
√(-3) = √(x²)
±√3i = x
Therefore, the solutions for x are:
x = ±√3i
The complex roots can be written in the form a + bi:
x = 1/4 (1 ± i√23)
The sum of the roots is:
1/4 (1 + i√23) + 1/4 (1 - i√23) = 1/2
The product of the roots is:
1/4 (1 + i√23) * 1/4 (1 - i√23) = 3/2
So, the solutions to the equation 2x - 6 = 4x² are:
x = 1/4 (1 - i√23)
x = 1/4 (1 + i√23)
where i is the imaginary unit (√(-1))[1].
I’m sorry to hear that you’ve been struggling with this quadratic equation! Let’s work through it together.
The given equation is: 2x−6=4x2
To find the value of (x), we can follow these steps:
1. Rearrange the equation to get all terms on one side:
[2x - 4x^2 - 6 = 0]
2. Combine like terms:
[4x^2 - 2x - 6 = 0]
3. Now, we can use the quadratic formula to find the solutions for (x):
[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}]
Here, (a = 4), (b = -2), and (c = -6).
Plugging in the values:
[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 4 \cdot (-6)}}{2 \cdot 4}]
Calculating the discriminant:
[b^2 - 4ac = (-2)^2 - 4 \cdot 4 \cdot (-6) = 4 + 96 = 100]
Taking the square root of the discriminant:
[\sqrt{100} = 10]
Now we have two possible solutions:
[x_1 = \frac{-(-2) + 10}{8} = \frac{12}{8} = 1.5]
[x_2 = \frac{-(-2) - 10}{8} = \frac{-12}{8} = -1.5]
So the solutions for the equation are (x = 1.5) and (x = -1.5). Feel free to double-check the calculations, and let me know if you have any other questions!
and I hope you got rid of that math sick !